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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 31

  • question_answer
    If \[{{S}_{n}}{{=}^{n}}{{C}_{0}}{{.}^{n}}{{C}_{n}}{{+}^{n}}{{C}_{1}}{{.}^{n}}{{C}_{n-1}}{{+}^{n}}{{C}_{2}}{{.}^{n}}{{C}_{n-2}}+....{{+}^{n}}{{C}_{n}}{{.}^{n}}{{C}_{0}},\]  then maximum value of \[\left[ \frac{{{S}_{n+1}}}{{{S}_{n}}} \right]\] is (where \[[.]\] denotes the greatest integer function)

    A) \[1\]                       

    B)        \[2\]                        

    C) \[3\]                     

    D)        \[4\]

    Correct Answer: C

    Solution :

      [c] \[{{S}_{n}}{{=}^{n}}{{C}_{0}}{{.}^{n}}{{C}_{n}}{{+}^{n}}{{C}_{1}}{{.}^{n}}{{C}_{n-1}}{{+}^{n}}{{C}_{2}}{{.}^{n}}{{C}_{n-2}}+....\]                                     \[{{+}^{n}}{{C}_{n}}{{.}^{n}}{{C}_{0}}\] \[{{=}^{n}}{{C}_{0}}{{.}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{.}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}{{.}^{n}}{{C}_{2}}+.....{{+}^{n}}{{C}_{n}}{{.}^{n}}{{C}_{n}}\] \[={{2}^{n}}{{C}_{n}}\] \[\Rightarrow \,\,\,\frac{{{S}_{n+1}}}{{{S}_{n}}}=\frac{^{2n+2}{{C}_{n+1}}}{^{2n}{{C}_{n}}}=\frac{(2n+2)\,(2n+1)}{(n+1)(n+1)}\]             \[=\frac{2(2n+1)}{n+1}=4-\frac{2}{n+1}\] \[\therefore \,\,\,\,{{\left[ \frac{{{S}_{n+1}}}{{{S}_{n}}} \right]}_{\max }}=3\]


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